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hw11-3 - 254 4 Interest Rate Derivative Securities T V(r T...

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254 4 Interest Rate Derivative Securities V ( r, T + ) - V ( r, T - ) = Z T + T - ∂V ∂t dt = - Z T + T - δ ( t - T ) dt = - 1 . That is, V ( r, T - ) = V ( r, T + ) + 1 = 1 . In the first problem V ( r, T - ) = V ( r, T + ) is also identically equal to one. Consequently, from T - to any t < T , the solutions of the two problems are the same. Actually the second problem can be understood as another form of the first problem. 14. Consider the problem: ∂V s 1 ∂t + 1 2 w 2 2 V s 1 ∂r 2 + ( u - λw ) ∂V s 1 ∂r - rV s 1 + 2 N k =1 δ ( t - T - k/ 2) = 0 , r l r r u , t T + N, V s 1 ( r, T + N ) = 0 , r l r r u . Show that V s 1 ( r, T ) gives the sum of values of 2 N zero-coupon bonds with maturities 1 / 2 , 1 , 3 / 2 , · · · , N years. Solution : The solution of the given problem is the sum of the solutions of the fol- lowing 2 N problems ∂V s 1 k ∂t + 1 2 w 2 2 V s 1 k ∂r 2 + ( u - λw ) ∂V s 1 k ∂r - rV s 1 k + δ ( t - T - k/ 2) = 0 , r l r r u , t T + N, V s 1 k ( r, T + N ) = 0 , r l r r u , k = 1 , 2 , · · · , 2 N. It is clear that V s 1 k ( r, t ) = 0 for t ( T + k/ 2 , T + N ] and for t < T + k/ 2, V s 1 k ( r, t ) satisfies ∂V s 1 k ∂t + 1 2 w 2 2 V s 1 k ∂r 2 + ( u - λw ) ∂V s 1 k ∂r - rV s 1 k + δ ( t - T - k/ 2) = 0 , r l r r u , t T + k/ 2 , V s 1 k ( r, T + k/ 2) = 0 , r l r r u .
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4 Interest Rate Derivative Securities 255 In Problem 13 we have shown that this problem can be rewritten as ∂V s 1 k ∂t + 1 2 w 2 2 V s 1 k ∂r 2 + ( u - λw ) ∂V s 1 k ∂r - rV s 1 k = 0 r l r r u , t T + k/ 2 , V s 1 k ( r, T + k/ 2) = 1 , r l r r u .
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